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When a ray travels in a depthstratified medium,
Snell's parameter
is constant along the ray.
If the ray emerges at the surface,
we can measure the distance that it has traveled,
the time it took, and its apparent speed .
A wellknown estimate
for the earth velocity contains this apparent speed.

(18) 
To see where this velocity estimate comes from,
first notice that the stratified velocity can be parameterized
as a function of time and takeoff angle of a ray from the surface.

(19) 
The coordinate of the tip of a ray with Snell parameter is
the horizontal component of velocity integrated over time.

(20) 
Inserting this into equation (3.18)
and canceling we have

(21) 
which shows that the observed velocity is the ``rootmeansquare'' velocity.
When velocity varies with depth,
the traveltime curve is only roughly a hyperbola.
If we break the event into many short line segments where the
th segment has a slope and a midpoint
each segment gives a different
and we have the unwelcome chore of assembling the best model.
Instead, we can fit the observational data to the best fitting hyperbola
using a different velocity hyperbola for each apex,
in other words,
find so this equation
will best flatten the data in space.

(22) 
Differentiate with respect to at constant getting

(23) 
which confirms that the observed velocity
in equation (3.18),
is the same as no matter where you measure
on a hyperbola.
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